for all such functions and in fact for all meromorphic functions in the Eremenko-Lyubich class \(\mathcal {B}\)  that have finite order and
for which \(\infty \)  is an asymptotic value. Bergweiler, Karpińska, and Stallard showed in [1]} that this result
even also holds for some functions \(f\in \mathcal {B}\)  having infinite order.
The situation is totally different when \(\infty \)  is not an asymptotical value as was shown by Bergweiler and Kotus in [2]} who, improving former results for elliptic functions by Kotus and Urbański (see [3]}, comp. [4]}), obtained the following upper estimate.
